Chapter 5

1. Chapter 5 Routing Algorithm in Networks

2. Routing Algorithm in Networks • How are message routed from origin to destination? • Circuit-Switching → telephone net. Dedicated bandwidth (path) • Message-switching : using routing table share link bandwidth concept, 同一msg用同一path • Packet-switching : 不同路徑for every packet out of order.

3. Routing Algorithm in Networks • Virtual circuit • – msg delivered in the order transmitted • – sharing, i.e. no dedicated paths. • Implementation • centralized ─ • distributed 1 2 4 1 1 1 2 1 3 6 3 5 6 1 5 4 1 Out of order Re-assembly problem Vulnerable to failure Comm. Of control infomation

4. Shortest Path Algorithm -Graph • G = (V, A) • dij is the src weight of (i, j) A. • dij =  if (i, j) A. • Source node is node 1 # of nodes |V|=N Set of nodes Set of Direct arcs

5. §5.2.3 Shortest Path • P.396 §Bellman-Ford Alg. (can handle negative weights but not negative cycles) • Let Di(h) be the length of a shortest path from 1 to i using h or fewer arcs (or links) • Initially D1(0) = 0 Di(0) = , i  1 • For each h = 0, 1, 2,…, N-2

7. §5.2.3 Shortest Path • [Thm] : The alg. Finds the correct shortest path lengths proof by induction. are the lengths of the shortest path using 1 or fewer links.

8. §5.2.3 Shortest Path • Induction step : suppose Di(h) are the correct length of shortest paths using h or fewer links. k1 dk1i dk2i k1 i dk3i k1

9. §5.2.3 Shortest Path • Complexity: • The alg. Requires at most N-1 iterations. • For each iteration, the recursion is performed by N-1 nodes. • Each application of the recursion requires no more than N-1 addition & comparisonsO(N3) complexity. Example : see P.397. Fig 5.31

10. §Dijkstra’s Algorithm.(position arc weights only.) • Let p be the set of nodes that are permanently labeled. • Step 0 : set P={1}, D1 = 0, and Dj = d1j, j  1 • Step 1 : Find i  P, s.t.

11. §Dijkstra’s Algorithm.(position arc weights only.) • Step 2 :

13. §Dijkstra’s Algorithm.(position arc weights only.) • After k iterations of the algorithm. P contains the k+1 nodes that are closest to node 1 and their permanent labels are the correct shortest path lengths from node 1. 1 2 4 1 1 2 1 6 3 5 1 2 1

14. §Dijkstra’s Algorithm.(position arc weights only.) • Proof by inductions: • True for k=1: • Inductive step: suppose statement is true for any k. P={1, i1,i2,…,ik} are (k+1) closest nodes to 1. Dk is length of the shortest path from 1 to k, kP.

15. §Dijkstra’s Algorithm.(position arc weights only.) ik-1 ik+1 i ik i1

16. §Dijkstra’s Algorithm.(position arc weights only.) • By construction, ik+1 is at least as close to 1 as any other node in • Complexity: • The algorithm requires N-1 iterations. • For each iteration: • Step 1 : requires at most N comparison • Step 2 : requires at most N addition & comparisons. complexity is O(N2)

17. §Floyd-Warshall algorithm • Finds shortest path between every pair of nodes O(N3) • Negative arc weights(but no negative cycle) Note: every pair Bellman-Ford O(N4) Dijkstra O(N3)

18. §Floyd-Warshall algorithm • Dij(n) = shortest path length between nodes I and j using only nodes 1,2,3…n as intermediate nodes on paths. Initially, Dij(0) = dij For n=0,1,2,…,N-1 Dij(n+1) = min[Dij(n+1) , Di,n+1(n+1)+dn+1,j ] for all i  j

20. §Floyd-Warshall algorithm • Example see fig.? • Proof by induction : • Complexity : • N iterations • Each iteration require N(N-1) comparisons & additions O(N3)

21. Centralized, synchronous Bellmen-Ford Algorithm • Let Di(h) be the shortest (<=h) path length from node 1 to node I • Initially, D1(h) = 0 for all h Di(0) =  for i = 1

22. Centralized, synchronous Bellmen-Ford Algorithm • => h is an index for iteration # (the # of links allowed in paths) ← synchronous • => 有困難 h 必須 all the same

23. § 5.2.4 Distributed Asynchronous Bellman-Ford Algorithm • Let Di be the shortest path length from node i to 1. • Let N(i) = { j|(i,j)  A }

24. § 5.2.4 Distributed Asynchronous Bellman-Ford Algorithm • Let Di be the shortest path length from node i to 1. • Let N(i) = { j|(i,j)  A } Neighbor • At each time t, each node i  1 has available: • Dji(t) : i’s latest estimate (sent by node i) of the shortest distance from node j  N(i) to node 1. • Di(t) : i’s latest estimate (computed by node i) of the shortest distance from node i to 1

25. § 5.2.4 Distributed Asynchronous Bellman-Ford Algorithm • At each point in time, each node. i  1 is doing one of the followings : • Node i update Di(t) by And leaves the estimate Dji(t), j  N(i) unchanged & sends Di, j  N(i) • Node i receives from one or more neighbors Di , j  N(i) computed by j at an earlier time. Node i update Dji(t) and leaves other estimate unchanged. • Node i is idle.

26. § 5.2.4 Distributed Asynchronous Bellman-Ford Algorithm • Prove convergence of the distributed asyn. B-F Alg. • Let Ti be the set of times at which node i update Di(t) • Let Tji be the set of times at which node i update Dj(t) • Let {t0, t1…} be the ordering of

27. § 5.2.4 Distributed Asynchronous Bellman-Ford Algorithm • Assumptions: • nodes never stop updateing their estimate Di(t) and receiving msg. from all their neighbors Dji(t), j  N(i), Ti, Tji have  # of elements. • All estimates Di(t), Dji(t), i  V, j  N(i) are non-negative. • Old distance information is eventually purged from the system.

28. § 5.2.4 Distributed Asynchronous Bellman-Ford Algorithm • Let Dik be the k-th iteration of Bellman-Ford (k=0,1,2…) for node i = 1,2,…N when • Let Dik be the k-th iteration of Bellman-Ford when Di(0) = 0, i = 1,2,…N

29. § 5.2.4 Distributed Asynchronous Bellman-Ford Algorithm

30. § 5.2.4 Distributed Asynchronous Bellman-Ford Algorithm

31. § 5.2.4 Distributed Asynchronous Bellman-Ford Algorithm

32. § 5.2.4 Distributed Asynchronous Bellman-Ford Algorithm